The behavior on fundamental group of a free pro-homotopy equivalence. II
type: Item
instance of: scholarly article
title: The behavior on fundamental group of a free pro-homotopy equivalence. II
zbMATH Open document ID: 0594.55008
DOI: 10.1016/0166-8641(86)90028-3
author: J. Dydak
author: Ross Geoghegan
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: In an earlier paper [ibid. 13, 239-253 (1982; Zbl 0481.55007)], the authors studied the question whether a morphism in pro- (pointed homotopy), which is an isomorphism in pro- (homotopy), is also an isomorphism in pro- (pointed homotopy). An affirmative answer to this question was given under suitable hypotheses. The present note continues this study and it is proved that the answer to this question is again affirmative provided the \(pro\)-\(\pi\) \({}_ 1\) of the target has pro- finite index. More specifically, the following is shown: If f: (X,x)\(\to (Y,y)\) is a morphism of pointed shape of pointed compacta such that \(f: X\to Y\) is an unpointed shape isomorphism, then f is a pointed shape isomorphism provided \(pro\)-\(\pi\) \({}_ 1(Y,y)\) has pro-finite index. The notion of pro-finite index is discussed.
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 55Q07
zbMATH DE Number: 3955982
zbMATH Keywords: pro-homotopy
zbMATH Keywords: pro-pointed homotopy
zbMATH Keywords: pro-fundamental group
zbMATH Keywords: pro-finite index
reviewed by: Sukhjit Singh
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90028-3
OpenAlex ID: W4250810168
cites work: The behavior on fundamental group of a free pro-homotopy equivalence
cites work: Detecting homotopy equivalences in base-point-free homotopy
cites work: Q3928677
cites work: Q3959112
description: scientific article; zbMATH DE number 3955982